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Legendre Polynomials are a system of complete and orthogonal polynomials, solutions to Legendre's differential equation. They are widely used in physics and engineering, particularly in problems involving spherical symmetry.
These functions generalize the Legendre polynomials and are solutions to the associated Legendre differential equation. They depend on two integer parameters, \(n\) (degree) and \(m\) (order).
Hermite Polynomials are a classical orthogonal polynomial sequence that arise as solutions to Hermite's differential equation. They form a complete basis for functions square-integrable with respect to a Gaussian weight function.
Laguerre Polynomials are solutions to Laguerre's differential equation. The associated Laguerre polynomials (often denoted \(L_n^k(x)\)) are particularly important in quantum mechanics.
Chebyshev Polynomials of the First Kind, denoted \(T_n(x)\), are a sequence of orthogonal polynomials related to trigonometric functions. They are defined on the interval [-1, 1].
Bessel functions of the first kind, \(J_n(x)\), are solutions to Bessel's differential equation that are finite at the origin \(x=0\). They arise naturally in problems involving cylindrical or spherical symmetry.
Bessel functions of the second kind, \(Y_n(x)\) (also known as Neumann functions), are a second set of linearly independent solutions to Bessel's differential equation. They are singular (diverge) at the origin \(x=0\).
Modified Bessel functions of the first kind, \(I_n(x)\), are solutions to the modified Bessel differential equation. Unlike \(J_n(x)\), they do not oscillate but grow exponentially for large \(x\). They are related to \(J_n(ix)\).
Modified Bessel functions of the second kind, \(K_n(x)\) (also known as Macdonald functions), are the second linearly independent solution to the modified Bessel equation. They decay exponentially for large \(x\).
Spherical Harmonics, \(Y_l^m(\theta, \phi)\), form a complete set of orthogonal functions on the surface of a sphere. They are the angular portion of the solution to Laplace's equation in spherical coordinates.
Spherical Bessel functions of the first kind, \(j_l(x)\), are related to the ordinary Bessel functions \(J_n(x)\) with half-integer order. They are solutions to the radial part of the Helmholtz equation in spherical coordinates and are regular at the origin.
Spherical Bessel functions of the second kind, \(y_l(x)\) (or Neumann functions \(n_l(x)\)), are related to the ordinary Bessel functions \(Y_n(x)\) with half-integer order. They are also solutions to the radial Helmholtz equation but are singular at the origin.
The Gamma function, \(\Gamma(x)\), is a generalization of the factorial function to complex and real numbers. It is defined via a convergent improper integral for \( \mathrm{Re}(x) > 0 \).
The Beta function, \(B(x, y)\), is a special function closely related to the Gamma function. It is defined by an integral over the interval [0, 1].
The Digamma function, \(\psi(x)\), is defined as the logarithmic derivative of the Gamma function. It is the first of the polygamma functions.
The (Gaussian or ordinary) Hypergeometric function, \({}_2F_1(a,b;c;z)\), is a special function represented by the hypergeometric series. Many other special functions are specific cases or related to it. It is a solution to the hypergeometric differential equation.
Elliptic integrals originally arose in connection with the problem of finding the arc length of an ellipse. The incomplete and complete elliptic integrals of the first (\(F\), \(K\)) and second (\(E\)) kind are standard forms.
Jacobi elliptic functions (\(\mathrm{sn}, \mathrm{cn}, \mathrm{dn}\)) are standard forms of elliptic functions that generalize trigonometric functions. They are doubly periodic functions in the complex plane and are related to elliptic integrals.
Airy functions, \(\mathrm{Ai}(x)\) and \(\mathrm{Bi}(x)\), are solutions to the Airy differential equation \(y'' - xy = 0\). \(\mathrm{Ai}(x)\) decays exponentially for \(x > 0\) and oscillates for \(x < 0\), while \(\mathrm{Bi}(x)\) grows exponentially for \(x > 0\).
The error function, \(\mathrm{erf}(x)\), is related to the cumulative distribution function of the normal (Gaussian) distribution. It represents the probability for a normally distributed random variable to fall within the interval \([-x\sigma\sqrt{2}, x\sigma\sqrt{2}]\). The complementary error function is \(\mathrm{erfc}(x) = 1 - \mathrm{erf}(x)\).
The Heaviside step function, \(H(x)\) or \(\theta(x)\), is a discontinuous function whose value is zero for negative arguments and one for non-negative arguments. It acts like a mathematical "switch".
The Dirac delta function, \(\delta(x)\), is not a true function but a distribution or generalized function. It is zero everywhere except at \(x=0\), where it is infinitely high, and its integral over the entire real line is equal to one. It represents an idealized point source or impulse.